Theorem ntrclsnvobr 40408

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then they are related the opposite way. (Contributed by RP, 21-May-2021.)

Hypotheses

Ref	Expression
ntrcls.o	⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)

Assertion

Ref	Expression
ntrclsnvobr	⊢ (𝜑 → 𝐾𝐷𝐼)

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝜑,𝑖,𝑗,𝑘

Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝐼(𝑖,𝑗,𝑘)   𝐾(𝑖,𝑗,𝑘)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsnvobr

Step	Hyp	Ref	Expression
1		ntrcls.o	. . 3 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . 3 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . 4 ⊢ (𝜑 → 𝐼𝐷𝐾)
4	2, 3	ntrclsbex 40390	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
5	1, 2, 4	dssmapnvod 40372	. 2 ⊢ (𝜑 → ◡𝐷 = 𝐷)
6	1, 2, 3	ntrclsf1o 40407	. . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
7		f1orel 6621	. . . 4 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → Rel 𝐷)
8		relbrcnvg 5971	. . . 4 ⊢ (Rel 𝐷 → (𝐾◡𝐷𝐼 ↔ 𝐼𝐷𝐾))
9	6, 7, 8	3syl 18	. . 3 ⊢ (𝜑 → (𝐾◡𝐷𝐼 ↔ 𝐼𝐷𝐾))
10	3, 9	mpbird 259	. 2 ⊢ (𝜑 → 𝐾◡𝐷𝐼)
11	5, 10	breqdi 5084	1 ⊢ (𝜑 → 𝐾𝐷𝐼)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536  Vcvv 3497   ∖ cdif 3936  𝒫 cpw 4542   class class class wbr 5069   ↦ cmpt 5149  ^◡ccnv 5557  Rel wrel 5563  –1-1-onto→wf1o 6357  ‘cfv 6358  (class class class)co 7159   ↑_m cmap 8409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-map 8411

This theorem is referenced by:  ntrclskex  40410  ntrclsfv2  40412  ntrclselnel2  40414  ntrclsfveq2  40417  ntrclsk4  40428